A Turbine

In a hydroelectric plant, water flowing over a dam drives a turbine, which runs a generator to make electric power. The figure shows a simplified physical model of the water hitting the turbine, in which it is assumed that the stream of water comes in at a 45 °angle with respect to the turbine blade, and bounces off at a 90 °angle at nearly the same speed. The water flows at a rate R, in units of kg/s, and the speed of the water is v. What are the magnitude and direction of the water's force on the turbine?

In a time interval Δt, the mass of water that strikes the blade is RΔt, and the magnitude of its initial momentum is mv = vRΔt. The water's final momentum vector is of the same magnitude, but in the perpendicular direction. By Newton's third law, the water's force on the blade is equal and opposite to the blade's force on the water. Since the force is constant, we can use the equation

Choosing the x axis to be to the right and the y axis to be up, this can be broken down into components as

and

The water's force on the blade thus has components

F_{water on blade,x} = vR

F_{water on blade,y} = -vR .

In situations like this, it is always a good idea to check that the result makes sense physically. The x component of the water's force on the blade is positive, which is correct since we know the blade will be pushed to the right. The y component is negative, which also makes sense because the water must push the blade down. The magnitude of the water's force on the blade is

and its direction is at a 45-degree angle down and to the right.